График функции
-1.0 -0.8 -0.6 -0.4 -0.2 1.0 0.0 0.2 0.4 0.6 0.8 5 -10
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0log ( ∣ tan ( x ) ∣ ) = 0 \log{\left (\left|{\tan{\left (x \right )}}\right| \right )} = 0 log ( ∣ tan ( x ) ∣ ) = 0 Решаем это уравнение Аналитическое решение x 1 = − π 4 x_{1} = - \frac{\pi}{4} x 1 = − 4 π x 2 = π 4 x_{2} = \frac{\pi}{4} x 2 = 4 π Численное решение x 1 = − 77.7544181763 x_{1} = -77.7544181763 x 1 = − 77.7544181763 x 2 = 90.3207887907 x_{2} = 90.3207887907 x 2 = 90.3207887907 x 3 = 22.7765467385 x_{3} = 22.7765467385 x 3 = 22.7765467385 x 4 = − 93.4623814443 x_{4} = -93.4623814443 x 4 = − 93.4623814443 x 5 = 77.7544181763 x_{5} = 77.7544181763 x 5 = 77.7544181763 x 6 = − 13.3517687778 x_{6} = -13.3517687778 x 6 = − 13.3517687778 x 7 = − 71.4712328692 x_{7} = -71.4712328692 x 7 = − 71.4712328692 x 8 = 33.7721210261 x_{8} = 33.7721210261 x 8 = 33.7721210261 x 9 = − 40.0553063333 x_{9} = -40.0553063333 x 9 = − 40.0553063333 x 10 = − 47.9092879672 x_{10} = -47.9092879672 x 10 = − 47.9092879672 x 11 = 66.7588438888 x_{11} = 66.7588438888 x 11 = 66.7588438888 x 12 = 69.9004365424 x_{12} = 69.9004365424 x 12 = 69.9004365424 x 13 = − 25.9181393921 x_{13} = -25.9181393921 x 13 = − 25.9181393921 x 14 = 63.6172512352 x_{14} = 63.6172512352 x 14 = 63.6172512352 x 15 = 30.6305283725 x_{15} = 30.6305283725 x 15 = 30.6305283725 x 16 = − 49.480084294 x_{16} = -49.480084294 x 16 = − 49.480084294 x 17 = 84.0376034835 x_{17} = 84.0376034835 x 17 = 84.0376034835 x 18 = − 87.1791961371 x_{18} = -87.1791961371 x 18 = − 87.1791961371 x 19 = 54.1924732744 x_{19} = 54.1924732744 x 19 = 54.1924732744 x 20 = 2.35619449019 x_{20} = 2.35619449019 x 20 = 2.35619449019 x 21 = − 33.7721210261 x_{21} = -33.7721210261 x 21 = − 33.7721210261 x 22 = 10.2101761242 x_{22} = 10.2101761242 x 22 = 10.2101761242 x 23 = 62.0464549084 x_{23} = 62.0464549084 x 23 = 62.0464549084 x 24 = 76.1836218496 x_{24} = 76.1836218496 x 24 = 76.1836218496 x 25 = 49.480084294 x_{25} = 49.480084294 x 25 = 49.480084294 x 26 = − 2.35619449019 x_{26} = -2.35619449019 x 26 = − 2.35619449019 x 27 = − 5.49778714378 x_{27} = -5.49778714378 x 27 = − 5.49778714378 x 28 = − 55.7632696012 x_{28} = -55.7632696012 x 28 = − 55.7632696012 x 29 = 60.4756585816 x_{29} = 60.4756585816 x 29 = 60.4756585816 x 30 = − 54.1924732744 x_{30} = -54.1924732744 x 30 = − 54.1924732744 x 31 = − 38.4845100065 x_{31} = -38.4845100065 x 31 = − 38.4845100065 x 32 = − 46.3384916404 x_{32} = -46.3384916404 x 32 = − 46.3384916404 x 33 = 40.0553063333 x_{33} = 40.0553063333 x 33 = 40.0553063333 x 34 = − 96.6039740979 x_{34} = -96.6039740979 x 34 = − 96.6039740979 x 35 = 41.6261026601 x_{35} = 41.6261026601 x 35 = 41.6261026601 x 36 = − 32.2013246993 x_{36} = -32.2013246993 x 36 = − 32.2013246993 x 37 = − 79.3252145031 x_{37} = -79.3252145031 x 37 = − 79.3252145031 x 38 = − 18.0641577581 x_{38} = -18.0641577581 x 38 = − 18.0641577581 x 39 = − 62.0464549084 x_{39} = -62.0464549084 x 39 = − 62.0464549084 x 40 = 44.7676953137 x_{40} = 44.7676953137 x 40 = 44.7676953137 x 41 = 46.3384916404 x_{41} = 46.3384916404 x 41 = 46.3384916404 x 42 = − 11.780972451 x_{42} = -11.780972451 x 42 = − 11.780972451 x 43 = 27.4889357189 x_{43} = 27.4889357189 x 43 = 27.4889357189 x 44 = − 65.188047562 x_{44} = -65.188047562 x 44 = − 65.188047562 x 45 = 71.4712328692 x_{45} = 71.4712328692 x 45 = 71.4712328692 x 46 = 85.6083998103 x_{46} = 85.6083998103 x 46 = 85.6083998103 x 47 = 32.2013246993 x_{47} = 32.2013246993 x 47 = 32.2013246993 x 48 = 74.6128255228 x_{48} = 74.6128255228 x 48 = 74.6128255228 x 49 = − 63.6172512352 x_{49} = -63.6172512352 x 49 = − 63.6172512352 x 50 = − 76.1836218496 x_{50} = -76.1836218496 x 50 = − 76.1836218496 x 51 = 18.0641577581 x_{51} = 18.0641577581 x 51 = 18.0641577581 x 52 = − 43.1968989869 x_{52} = -43.1968989869 x 52 = − 43.1968989869 x 53 = − 99.7455667515 x_{53} = -99.7455667515 x 53 = − 99.7455667515 x 54 = − 60.4756585816 x_{54} = -60.4756585816 x 54 = − 60.4756585816 x 55 = − 90.3207887907 x_{55} = -90.3207887907 x 55 = − 90.3207887907 x 56 = − 16.4933614313 x_{56} = -16.4933614313 x 56 = − 16.4933614313 x 57 = − 69.9004365424 x_{57} = -69.9004365424 x 57 = − 69.9004365424 x 58 = 88.7499924639 x_{58} = 88.7499924639 x 58 = 88.7499924639 x 59 = 3.92699081699 x_{59} = 3.92699081699 x 59 = 3.92699081699 x 60 = 93.4623814443 x_{60} = 93.4623814443 x 60 = 93.4623814443 x 61 = 11.780972451 x_{61} = 11.780972451 x 61 = 11.780972451 x 62 = 98.1747704247 x_{62} = 98.1747704247 x 62 = 98.1747704247 x 63 = − 19.6349540849 x_{63} = -19.6349540849 x 63 = − 19.6349540849 x 64 = 38.4845100065 x_{64} = 38.4845100065 x 64 = 38.4845100065 x 65 = − 21.2057504117 x_{65} = -21.2057504117 x 65 = − 21.2057504117 x 66 = 24.3473430653 x_{66} = 24.3473430653 x 66 = 24.3473430653 x 67 = − 84.0376034835 x_{67} = -84.0376034835 x 67 = − 84.0376034835 x 68 = − 35.3429173529 x_{68} = -35.3429173529 x 68 = − 35.3429173529 x 69 = − 41.6261026601 x_{69} = -41.6261026601 x 69 = − 41.6261026601 x 70 = − 30.6305283725 x_{70} = -30.6305283725 x 70 = − 30.6305283725 x 71 = − 91.8915851175 x_{71} = -91.8915851175 x 71 = − 91.8915851175 x 72 = 82.4668071567 x_{72} = 82.4668071567 x 72 = 82.4668071567 x 73 = 96.6039740979 x_{73} = 96.6039740979 x 73 = 96.6039740979 x 74 = 25.9181393921 x_{74} = 25.9181393921 x 74 = 25.9181393921 x 75 = − 27.4889357189 x_{75} = -27.4889357189 x 75 = − 27.4889357189 x 76 = − 24.3473430653 x_{76} = -24.3473430653 x 76 = − 24.3473430653 x 77 = − 8.63937979737 x_{77} = -8.63937979737 x 77 = − 8.63937979737 x 78 = − 82.4668071567 x_{78} = -82.4668071567 x 78 = − 82.4668071567 x 79 = − 10.2101761242 x_{79} = -10.2101761242 x 79 = − 10.2101761242 x 80 = − 74.6128255228 x_{80} = -74.6128255228 x 80 = − 74.6128255228 x 81 = 325.94023781 x_{81} = 325.94023781 x 81 = 325.94023781 x 82 = − 85.6083998103 x_{82} = -85.6083998103 x 82 = − 85.6083998103 x 83 = − 57.334065928 x_{83} = -57.334065928 x 83 = − 57.334065928 x 84 = − 98.1747704247 x_{84} = -98.1747704247 x 84 = − 98.1747704247 x 85 = 47.9092879672 x_{85} = 47.9092879672 x 85 = 47.9092879672 x 86 = 16.4933614313 x_{86} = 16.4933614313 x 86 = 16.4933614313 x 87 = − 3.92699081699 x_{87} = -3.92699081699 x 87 = − 3.92699081699 x 88 = 68.3296402156 x_{88} = 68.3296402156 x 88 = 68.3296402156 x 89 = 19.6349540849 x_{89} = 19.6349540849 x 89 = 19.6349540849 x 90 = 5.49778714378 x_{90} = 5.49778714378 x 90 = 5.49778714378 x 91 = 99.7455667515 x_{91} = 99.7455667515 x 91 = 99.7455667515 x 92 = 52.6216769476 x_{92} = 52.6216769476 x 92 = 52.6216769476 x 93 = − 52.6216769476 x_{93} = -52.6216769476 x 93 = − 52.6216769476 x 94 = − 68.3296402156 x_{94} = -68.3296402156 x 94 = − 68.3296402156 x 95 = 55.7632696012 x_{95} = 55.7632696012 x 95 = 55.7632696012 x 96 = 91.8915851175 x_{96} = 91.8915851175 x 96 = 91.8915851175 x 97 = 8.63937979737 x_{97} = 8.63937979737 x 97 = 8.63937979737
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:log ( ∣ tan ( 0 ) ∣ ) \log{\left (\left|{\tan{\left (0 \right )}}\right| \right )} log ( ∣ tan ( 0 ) ∣ ) f ( 0 ) = ∞ ~ f{\left (0 \right )} = \tilde{\infty} f ( 0 ) = ∞ ~
Экстремумы функции
Для того, чтобы найти экстремумы, нужно решить уравнениеd d x f ( x ) = 0 \frac{d}{d x} f{\left (x \right )} = 0 d x d f ( x ) = 0 d d x f ( x ) = \frac{d}{d x} f{\left (x \right )} = d x d f ( x ) = Первая производная sign ( tan ( x ) ) ∣ tan ( x ) ∣ ( tan 2 ( x ) + 1 ) = 0 \frac{\operatorname{sign}{\left (\tan{\left (x \right )} \right )}}{\left|{\tan{\left (x \right )}}\right|} \left(\tan^{2}{\left (x \right )} + 1\right) = 0 ∣ tan ( x ) ∣ sign ( tan ( x ) ) ( tan 2 ( x ) + 1 ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 (0, zoo) Интервалы возрастания и убывания функции:
Точки перегибов
Найдем точки перегибов, для этого надо решить уравнениеd 2 d x 2 f ( x ) = 0 \frac{d^{2}}{d x^{2}} f{\left (x \right )} = 0 d x 2 d 2 f ( x ) = 0 d 2 d x 2 f ( x ) = \frac{d^{2}}{d x^{2}} f{\left (x \right )} = d x 2 d 2 f ( x ) = Вторая производная ( tan 2 ( x ) + 1 ) ( 2 δ ( tan ( x ) ) ∣ tan ( x ) ∣ ( tan 2 ( x ) + 1 ) − sign 2 ( tan ( x ) ) tan 2 ( x ) ( tan 2 ( x ) + 1 ) + 2 sign ( tan ( x ) ) ∣ tan ( x ) ∣ tan ( x ) ) = 0 \left(\tan^{2}{\left (x \right )} + 1\right) \left(\frac{2 \delta\left(\tan{\left (x \right )}\right)}{\left|{\tan{\left (x \right )}}\right|} \left(\tan^{2}{\left (x \right )} + 1\right) - \frac{\operatorname{sign}^{2}{\left (\tan{\left (x \right )} \right )}}{\tan^{2}{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) + \frac{2 \operatorname{sign}{\left (\tan{\left (x \right )} \right )}}{\left|{\tan{\left (x \right )}}\right|} \tan{\left (x \right )}\right) = 0 ( tan 2 ( x ) + 1 ) ( ∣ tan ( x ) ∣ 2 δ ( tan ( x ) ) ( tan 2 ( x ) + 1 ) − tan 2 ( x ) sign 2 ( tan ( x ) ) ( tan 2 ( x ) + 1 ) + ∣ tan ( x ) ∣ 2 sign ( tan ( x ) ) tan ( x ) ) = 0 Решаем это уравнение x 1 = π 4 x_{1} = \frac{\pi}{4} x 1 = 4 π Интервалы выпуклости и вогнутости: [pi/4, oo) (-oo, pi/4]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-ooTrue Возьмём предел y = lim x → − ∞ log ( ∣ tan ( x ) ∣ ) y = \lim_{x \to -\infty} \log{\left (\left|{\tan{\left (x \right )}}\right| \right )} y = x → − ∞ lim log ( ∣ tan ( x ) ∣ ) True Возьмём предел y = lim x → ∞ log ( ∣ tan ( x ) ∣ ) y = \lim_{x \to \infty} \log{\left (\left|{\tan{\left (x \right )}}\right| \right )} y = x → ∞ lim log ( ∣ tan ( x ) ∣ )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции log(Abs(tan(x))), делённой на x при x->+oo и x ->-ooTrue Возьмём предел y = x lim x → − ∞ ( 1 x log ( ∣ tan ( x ) ∣ ) ) y = x \lim_{x \to -\infty}\left(\frac{1}{x} \log{\left (\left|{\tan{\left (x \right )}}\right| \right )}\right) y = x x → − ∞ lim ( x 1 log ( ∣ tan ( x ) ∣ ) ) True Возьмём предел y = x lim x → ∞ ( 1 x log ( ∣ tan ( x ) ∣ ) ) y = x \lim_{x \to \infty}\left(\frac{1}{x} \log{\left (\left|{\tan{\left (x \right )}}\right| \right )}\right) y = x x → ∞ lim ( x 1 log ( ∣ tan ( x ) ∣ ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).log ( ∣ tan ( x ) ∣ ) = log ( ∣ tan ( x ) ∣ ) \log{\left (\left|{\tan{\left (x \right )}}\right| \right )} = \log{\left (\left|{\tan{\left (x \right )}}\right| \right )} log ( ∣ tan ( x ) ∣ ) = log ( ∣ tan ( x ) ∣ ) log ( ∣ tan ( x ) ∣ ) = − log ( ∣ tan ( x ) ∣ ) \log{\left (\left|{\tan{\left (x \right )}}\right| \right )} = - \log{\left (\left|{\tan{\left (x \right )}}\right| \right )} log ( ∣ tan ( x ) ∣ ) = − log ( ∣ tan ( x ) ∣ ) 